616 Mr. R. Hargreaves on Wien's Law. 



Note : — When m=0 the solution of (1) is 

 t = exp. (-JP . dx)xi±^ 2 + r2 -v, 

 and when r = the solution is 



t=x* exp. ( — J i 3 . dx)^2p[m \~— — ) ' 

 January, 1914. 



LXIX. On Wiens Laic. By R. Hargreaves, M.A.^ 



WIEN'S Law has been approached in a variety of ways, 

 but the arguments usually lack some element of 

 generality as regards the shape of enclosure or type of 

 motion. It is proposed to deal here with the mean effect for 

 an enclosure of any shape, in which each element dS of 

 surface has an arbitrary normal component w of small motion. 

 In the element of time dt a change of wave-length affects 

 the radiation in a small part of the whole volume pro- 

 portional to dt, and this is exactly representative of the 

 action to which radiation from every part of the interior will 



in turn be exposed. The primary effect -r-^ = - — ^ — ■, with 



i for angle of incidence, is to be modified by taking into 

 account the ratio of the volume affected in time dt to the 

 total volume, and also a mean is to be taken for various 

 angles of incidence. The resulting mean is 



A\ C Cz 2w cos i dSVdt cos i . . ,. fwdtdS Ar /nN 



Here d$ . Vdt cos i is the volume from which radiation 

 reaches the surface in time dt at angle i, and half of this 

 volume is taken because only the half of the radiation which 

 is approaching dS is affected by the change. The factor 

 sin i di is used to average the effects for various angles of 

 incidence, and as the integral of sin i di is 1, it is not necessary 

 to write this integral in the denominator. 



* Tables of this function appear in Gray & Matthews's 'Bessel 

 Functions/ and the Bessel function of the second kind Y furnishing the 

 second solution in this case has been tabulated by B. A. Smith (Messenger 

 of Mathematics, xxvi. p. 98). 



Two other known functions, \e~^ . dx (the probability integral), and 



(V 2 . dx, tabulated by H. G. Dawson (Proc. Lond. Math. Soc. xxix. 



p. 519, 1898), occur when 4n is an odd integer. [See (25).] 

 t Communicated by the Author. 



